Study of ${e}^{+}{e}^{-}\to {\pi }^{0}X\left(3872\right)\gamma $ and search for ${Z}_{c}\left(4020{\right)}^{0}\to X\left(3872\right)\gamma $

Using data samples collected with the BESIII detector operating at the BEPCII storage ring at center-of-mass energies from 4.178 to 4.600 GeV, we study the process ${e}^{+}{e}^{-}\to {\pi }^{0}X\left(3872\right)\gamma $ and search for ${Z}_{c}\left(4020{\right)}^{0}\to X\left(3872\right)\gamma $. We find no significant signal and set upper limits on $\sigma \left({e}^{+}{e}^{-}\to {\pi }^{0}X\left(3872\right)\gamma \right)·B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi \right)$ and $\sigma \left({e}^{+}{e}^{-}\to {\pi }^{0}{Z}_{c}\left(4020{\right)}^{0}\right)·B\left({Z}_{c}\left(4020{\right)}^{0}\to X\left(3872\right)\gamma \right)·B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi \right)$ for each energy point at 90% confidence level, which is of the order of several tenths pb.